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24
Analysis and Geometry on Configuration Spaces
, 1997
"... In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X ..."
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Cited by 50 (8 self)
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In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X. This geometry is "nonflat" even if X = IR d . It is obtained as a natural lifting of the Riemannian structure on X. In particular, a corresponding gradient r \Gamma , divergence div \Gamma , and LaplaceBeltrami operator H \Gamma = \Gammadiv \Gamma r \Gamma are constructed. The associated volume elements, i.e., all measures ¯ on \Gamma X w.r.t. which r \Gamma and div \Gamma become dual operators on L 2 (\Gamma X ; ¯), are identified as exactly the mixed Poisson measures with mean measure equal to a multiple of the volume element dx on X. In particular, all these measures obey an integration by parts formula w.r.t. vector fields on \Gamma X . The corresponding Dirichlet...
Drift transforms and Green function estimates for discontinuous processes
 JOURNAL OF FUNCTIONAL ANALYSIS
, 2003
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Construction of Diffusions on Configuration Spaces
"... We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Ga ..."
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Cited by 27 (3 self)
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We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Gamma E ; ¯) for a large class of measures ¯ on \Gamma E (without assuming an integration by parts formula) generalizing all corresponding results known so far. Subsequently, we prove that under mild conditions the Dirichlet forms E \Gamma ¯ are quasiregular, and that hence there exist associated diffusions on \Gamma E , provided E is a complete separable metric space and \Gamma E is equipped with a suitable topology, which is the vague topology if E is locally compact. We discuss applications to the case where E is a finite dimensional manifold yielding an existence result on diffusions on \Gamma E which was already announced in [AKR96a, AKR96b], resp. used in [AKR98, AKR97b]. Furthermore...
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 23 (6 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
Intrinsic ultracontractivity of the Feynman–Kac semigroup for relativisitic stable processes
 ZBL 1112.47034 MR 2231884
, 2006
"... Let Xt be the relativistic αstable process in Rd, α ∈ (0, 2), d>α, with infinitesimal generator H (α) 0 = −((− ∆ +m2/α) α/2 − m). We study intrinsic ultracontractivity (IU) for the FeynmanKac semigroup Tt for this process with generator H (α) 0 − V, V ≥ 0, V locally bounded. We prove that if lim ..."
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Cited by 21 (3 self)
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Let Xt be the relativistic αstable process in Rd, α ∈ (0, 2), d>α, with infinitesimal generator H (α) 0 = −((− ∆ +m2/α) α/2 − m). We study intrinsic ultracontractivity (IU) for the FeynmanKac semigroup Tt for this process with generator H (α) 0 − V, V ≥ 0, V locally bounded. We prove that if lim x→ ∞ V (x) =∞, then for every t>0 the operator Tt is compact. We consider the class V of potentials V such that V ≥ 0, lim x→ ∞ V (x) = ∞ and V is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For V in the class V we show that the semigroup Tt is IU if and only if lim x→ ∞ V (x)/x  = ∞. If this condition is satisfied we also obtain sharp estimates of the first eigenfunction φ1 for Tt. Inparticular, when V (x) =x  β, β>0, then the semigroup Tt is IU if and only if β>1. For β>1 the first eigenfunction φ1(x) is comparable to exp(−m 1/α x)(x  +1) (−d−α−2β−1)/2.
Riesz transform and L p cohomology for manifolds with Euclidean ends
 Duke Math. J
"... Abstract. Let M be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, R n \ B(0, R) for some R> 0, each of which carries the standard metric. Our main result is that the Riesz transform on M is bounded from L p (M) → L p (M; T ∗ M) for 1 < p < n ..."
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Cited by 8 (2 self)
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Abstract. Let M be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, R n \ B(0, R) for some R> 0, each of which carries the standard metric. Our main result is that the Riesz transform on M is bounded from L p (M) → L p (M; T ∗ M) for 1 < p < n and unbounded for p ≥ n if there is more than one end. It follows from known results that in such a case the Riesz transform on M is bounded for 1 < p ≤ 2 and unbounded for p> n; the result is new for 2 < p ≤ n. We also give some heat kernel estimates on such manifolds. We then consider the implications of boundedness of the Riesz transform in L p for some p> 2 for a more general class of manifolds. Assume that M is a ndimensional complete manifold satisfying the Nash inequality and with an O(r n) upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on L p for some p> 2 implies a Hodgede Rham interpretation of the L p cohomology in degree 1, and that the map from L 2 to L p cohomology in this degree is injective. 1.
Interpolation of Sobolev spaces, LittlewoodPaley inequalities and Riesz transforms on graphs
 PUBLICACIONS MATEMATIQUES
"... Abstract. Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(x) = ∑ y p(x, y)f(y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ‖∇f‖ p and ∥ ∥ (I − P) ..."
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Cited by 5 (2 self)
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Abstract. Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(x) = ∑ y p(x, y)f(y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ‖∇f‖ p and ∥ ∥ (I − P)
Path Integral Representation for Schrödinger Operators with Bernstein Functions of the Laplacian
"... Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The onetoone correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman ..."
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Cited by 5 (4 self)
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Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The onetoone correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard FeynmanKac formula is taken here by subordinated Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on selfadjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and halfinteger spin values. This approach also allows to propose a notion of generalized Kato class for which hypercontractivity of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived. 1 2 1
The hypergroup property and representation of Markov Kernels
 Séminare de Probabilitiés XLI, Lecture notes in Mathematics, Vol
, 2006
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